{"title":"Optimal conditions with chattering in the inverted two-link pendulum control problem","authors":"M.N. Ronzhina","doi":"10.1016/j.jappmathmech.2016.05.004","DOIUrl":null,"url":null,"abstract":"<div><p><span>The plane motion of a two-link inverted mathematical pendulum, attached by a hinge to a moving trolley, is studied. The pendulum is controlled by a bounded force applied to the trolley. The problem of the minimization of the mean square<span> deviation of the pendulum from an unstable equilibrium position is considered. </span></span>Pontryagin's maximum principle<span><span> is used. An optimal feedback control, containing special second order trajectories and trajectories with chattering is constructed for a linearized model. It is proved that, before emerging onto a special manifold, the optimal trajectories experience a chattering after a finite </span>period of time<span> and then reach the unstable equilibrium after an infinite time by a specific mode. The global optimality of the solution constructed is proved.</span></span></p></div>","PeriodicalId":49686,"journal":{"name":"Pmm Journal of Applied Mathematics and Mechanics","volume":"80 1","pages":"Pages 16-23"},"PeriodicalIF":0.0000,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.jappmathmech.2016.05.004","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Pmm Journal of Applied Mathematics and Mechanics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021892816300417","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 3
Abstract
The plane motion of a two-link inverted mathematical pendulum, attached by a hinge to a moving trolley, is studied. The pendulum is controlled by a bounded force applied to the trolley. The problem of the minimization of the mean square deviation of the pendulum from an unstable equilibrium position is considered. Pontryagin's maximum principle is used. An optimal feedback control, containing special second order trajectories and trajectories with chattering is constructed for a linearized model. It is proved that, before emerging onto a special manifold, the optimal trajectories experience a chattering after a finite period of time and then reach the unstable equilibrium after an infinite time by a specific mode. The global optimality of the solution constructed is proved.
期刊介绍：
This journal is a cover to cover translation of the Russian journal Prikladnaya Matematika i Mekhanika, published by the Russian Academy of Sciences and reflecting all the major achievements of the Russian School of Mechanics.The journal is concerned with high-level mathematical investigations of modern physical and mechanical problems and reports current progress in this field. Special emphasis is placed on aeronautics and space science and such subjects as continuum mechanics, theory of elasticity, and mathematics of space flight guidance and control.
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